Modeling of Transport, Chemical and Electrochemical Processes in Solid Oxide Fuel Cells Thinh Xuan Ho Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen 2009 Acknowledgements First and foremost I would like to thank my supervisor, Professor Alex C. Hoffmann from the Department of Physics and Technology for his continuous support and kindness during my time being with his group. He accepted me into his group and that enabled me to go to Norway and perform this thesis. Moreover, he was always beside me and even sometimes pushed me to go ahead when I met difficulties by his friendly guidance. I would also like to thank Dr. Pawel Kosinski, my co-supervisor, for his useful guidance. He has also been very kind to me right from the beginning when I first came to Norway. I am highly indebted to the staff at Prototech for accepting me as a PhD student and always being kind to me during my time with them. I would especially like to thank Mr. Arild Vik, the technical director of Prototech AS, who brought me to a very interesting world of fuel cells and gave me a lot freedom in doing the PhD project. Many thanks go to Tor Monsen, amongst many others, who forced me to speak Norwegian but was always kind to me even I could not. My sincere gratitude goes to the Department of Physics and Technology for accepting my enrollment for PhD studies and offering me excellent working conditions, even though most of the work was carried out at Prototech AS. My friends and colleagues within the group of Multiphase Process at the Department are highly appreciated for exchanging experiences and ideas with me, especially during our CFD meetings, and for creating a highly supportive working environment and also for joyful moments over the last four years that I have had in Norway. I would like to thank my parents, my grandmother and my father-in-law for their love and mental support. Finally, I want to thank my wife, Loan, and my son, Vinh for being with me. My wife has given up her job in Vietnam to be with me and shared with me very many 3 things. My wife and my son were always supportive to me even during my most stressful time! Without their love, patience and understanding, I definitely could not finish this thesis. Thank you very much Loan and Vinh! Contents 1 Organization of the thesis 9 1.1 Papers included in the thesis . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Papers not included in the thesis . . . . . . . . . . . . . . . . . . . . 10 2 General Introduction 11 2.1 Solid oxide fuel cells . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Chemical and electrochemical reactions . . . . . . . . . . . . . 14 2.1.2 The electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3 The electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.4 The interconnect . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Aim of the current study . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Modeling of Solid Oxide Fuel Cells: Review 21 3.1 Modeling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.1 Cell-component level . . . . . . . . . . . . . . . . . . . . . . . 22 3.1.2 Cell and/or stack level . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Heat sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.1 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Heat of chemical and electrochemical reactions . . . . . . . . . 30 3.2.3 Joule heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 4 Summary of papers included in the thesis 33 4.1 Paper 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Paper 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3 Paper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Paper 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 Paper 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 Paper 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 Concluding remarks and further work 37 6 Abstract The working of solid oxide fuel cells (SOFCs) involve fluid dynamics, chemical re- actions and electrochemical processes. These phenomena happen simultaneously in complex and sophisticated structures of the SOFC main components consisting of gas channels, porous electrodes, dense electrolyte and interconnects. Therefore, modeling of SOFCs with consideration of the detailed processes, which is indispens- ably important in the development of the fuel cells, is not always an easy task. The chemical reactions include the steam reforming of methane and the water– gas shift reaction. The former occurs heterogeneously on the anode surface and homogeneously in the fuel channel while the later occurs homogeneously everywhere in the anode compartment. The electrochemical reactions are oxidation of hydrogen and/or carbon monoxide and reduction of oxygen, which take place at the so-called ”three-phase boundaries” (TPBs) formed by the presence of all three of the electrode, the electrolyte and the gas phase. When ionic–electronic conducting composite electrodes are used, the TPBs extends from electrode–electrolyte interfaces into the electrodes forming an electrochemically active layer with finite thickness. A numerical model for the detailed processes happening in SOFCs is always needed. Advantage of a model is that it can provide detailed insights into the cells that can not be gained by experiments. Additionally, it helps investigating impacts of each process parameter and their interaction, giving information for cell optimiza- tion. Modeling of SOFCs has been increasing rapidly during the last two decades, especially the last few years. However, models considering detailed processes taking place at TPBs or considering effects of the composite electrodes are still relatively rare. This thesis develops a detailed numerical model for planar solid oxide fuel cells. In this model, the electrochemical reactions are assumed to take place in the elec- trochemically active (functional) layers of finite thickness. The thickness of these 7 functional layers is up to 50µm, and depends among other things on the size of the particles from which the electrodes are made. The heat of the electrochemical reactions is assumed to be released on the anode side. Moreover, steady-state electri- cal field-driven transport of electrons and oxygen-ions in the composite electrodes– electrolyte assembly are modeled using an algorithm for Fickian diffusion built into the commercial CFD package Star-CD. Moreover, in the developed model, one single computational domain includes the air and fuel channels, the electrodes–electrolyte assembly and/or the interconnects, and thus constitutes a single and continuous domain in which balances of mass, mo- mentum, chemical species and energy associated with chemical and electrochemical processes are solved simultaneously. The model is firstly applied to an anode-supported cell with co- and counter-flow configurations. The oxidation of carbon monoxide is included in this application, however, results show insignificant impact of it on performance of the cell. It is then applied to a cathode-supported cell, which showed a better performance in terms of temperature and current density distributions compared to the anode- supported design. In these applications, the computational domain does not include the interconnects and only variation along two directions (along the cell length and direction normal to the electrolyte surface) are captured. The model is then applied to fully three-dimensional modeling of an anode-supported cell. In this investigation, the interconnects are included, therefore, their effects on the cell performance are observed. In addition to the studies mentioned above, a discussion on transport of oxygen ions in the electrolyte is carried out. Some scenarios relating to ion fluxes are proposed, in which the Nernst–Planck and Poisson equations are solved for concentration of ions and potential distribution in the electrolyte. 8 Chapter 1 Organization of the thesis This thesis is written in a paper form, which consists of an introductory section fol- lowed by a section with scientific papers. The introductory part consists of chapters 2, 3, 4 and 5 while the scientific papers includes papers published or accepted for publication in international journals, papers presented at conferences, and technical reports which will be submitted for publication later on. In the introductory part, chapter 2 presents a relatively short introduction to solid oxide fuel cells (SOFCs). A brief description of state-of-the-art SOFC components is also given in this chapter. Chapter 3 gives a literature review on modeling of SOFCs. The papers, which are included in this thesis, will briefly be summarized in chapter 4. Finally, concluding remarks and further work are presented in Chapter 5. The following sections represent a list of papers that are included in the thesis and added after the introductory part. Papers which are not included in the thesis are named as well. 1.1 Papers included in the thesis 1. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2008. Numerical modeling of solid oxide fuel cells. Chemical Engineering Science 63 (21), 5356–5365. 2. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2008. Numerical study of an SOFC with direct internal reforming using charge diffusion-based model. Proceedings of The 8th European SOFC Forum, 30th June–4th July, Lucerne, Switzerland. 9 3. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2009. Numerical analysis of a planar anode-supported SOFC with composite electrodes. International Journal of Hydrogen Energy 34, 3488–3499. 4. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2009. Modeling of transport, chem- ical and electrochemical phenomena in a cathode-supported SOFC. Chemical Engineering Science, doi:10.1016/j.ces.2009.03.043. 5. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2009. Numerical modeling of SOFCs using a fully three-dimensional approach. Technical report. 6. Ho TX, Kosinski P, Hoffmann AC, 2009. Discussion on the transport of oxygen-ions in an SOFC electrolyte. Technical report. 1.2 Papers not included in the thesis 1. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2009. Fully three-dimensional mod- eling of solid oxide fuel cells. The Sixth Symposium on Fuel Cell Modelling and Experimental Validation, 25th –26th March, Bad Herrenalb, Karlsruhe, Ger- many (oral presentation). 2. Ho TX, Kosinski P, Hoffmann AC, Vik A, 2008. Numerical study of a SOFC with direct internal methane reforming. Norwegian Hydrogen Seminar, 25th – 26th September, Bergen, Norway (poster). 3. Ho TX, Kosinski P, Hoffmann AC, Wærnhus I, Vik A, 2007. Numerical sim- ulation of electrochemical and transport processes in solid oxide fuel cells. Proceedings of SOFC-X, The Tenth International Symposium on Solid Oxide Fuel Cells, 3rd –8th June, Nara, Japan (oral presentation). 4. Ho TX, Kosinski P, Hoffmann AC, 2006. Direct numerical simulation of particle-fluid flow: The state-of-the-art. Proceedings of WCPT5, The Fifth World Conference on Particle Technology, 23rd –27th April, Orlando, Florida, USA (oral presentation). 10 Chapter 2 General Introduction Fuel cells are energy conversion devices, which directly convert chemical energy into electrical energy with high efficiency, low pollution and low noise. The process of combining hydrogen and oxygen to generate water and electricity is indeed a reversion of an electrolysis process, and was found by William Grove during his experiment in 1839 [1]. A typical fuel cell consists of a dense electrolyte, porous anode and cathode electrodes, fuel and oxidant channels and interconnects as main components. Fuel is oxidized at the anode releasing electrons, which then transport to the cathode via an external circuit. At the cathode, oxidant (mostly oxygen) is reduced consuming the transported electrons. The electrolyte allows ions to flow through in order to complete the overall electrochemical reactions. Depending on the nature of the electrolyte used, fuel cells are categorized in different types. Figure 2.1 represents the working principles of different types of fuel cells. Common types of fuel cells currently under development include [1, 2]: – Alkaline fuel cells (AFCs), – Polymer electrolyte membrane fuel cells (PEMFCs), – Phosphoric acid fuel cells (PAFCs), – Direct methanol fuel cells (DMFCs), – Molten carbonate fuel cells (MCFCs), – Solid oxide fuel cells (SOFCs). 11 Fuel e Oxidant e H2 OH - O2 AFC H2O H2O PEMFC H+ O2 H2 PAFC H2O CH3OH O2 DMFC H2O H+ CO2 H2O H2 O2 MCFC CO2- 3 H2O CO2 CO2 H2 O2- SOFC O2 H2O Anode Electrolyte Cathode Figure 2.1: Principle of different types of fuel cells. The first four types of fuel cells are known as low- and medium-temperature fuel cells, which operate at temperatures ranging from room temperature up to around 220◦ C. The last two types are high-temperature fuel cells operating at temperatures of 500–1000◦ C. These cells differ in many aspects such as their constituent materials, fuels, operating conditions and performance characteristics. Table 2.1 represents characteristics of different types of fuel cells. The focus of this thesis is on the last type of fuel cells, SOFCs, which is described in the following sections. Table 2.1: Types of fuel cells and their characteristics [1–4] Type Mobile ionOperating Application and notes temperature AFC OH− 50–220◦ C Military, space, e.g. Apollo and Shuttle + PEMFC H 30–100◦ C Vehicles and mobile application, low power CHP systems + ◦ PAFC H 150–220 C Large CHP systems DMFC H+ 20–90◦ C Portable electronic systems 2− ◦ MCFC CO3 600–700 C Medium to large CHP systems 2− SOFC O 500–1000◦ C Stationary electric power, small to large CHP systems CHP: Combined heat-and-power 12 2.1 Solid oxide fuel cells Solid oxide fuel cells (SOFCs) use a solid-ceramic electrolyte and operate at high temperatures (500–1000◦ C). The electrolyte allows oxygen ions to transport through its crystal lattice via available vacancies. SOFCs possess a number of interesting features due to their high operating temperature and have therefore been receiving worldwide attention during the last two decades. Solid oxide fuel cells may yield an electrical efficiency as high as 55%. Moreover, they are capable of working in hybrid systems with gas turbines and combined heat-and- power (CHP) generation, giving overal efficiencies up to 70% and 90%, respectively [5, 6]. Other advantages of SOFCs include: – The capability of working with a relatively wide range of fuels, i.e. hydrogen, methane or natural gas and hydrocarbons. – No expensive catalyst is needed for electrochemical reactions. – The solid nature of the electrolyte gives geometrical flexibility of cell designs; planar, tubular and monolithic designs are known. e- Anode interconnect _ Fuel channel e- Fuel Anode Load Electrolyte O2- O2- O2- O2- Cathode Air channel Air e- Cathode + interconnect Figure 2.2: Diagram of part of a planar solid oxide fuel cell. Figure 2.2 represents part of a planar solid oxide fuel cell. In the figure, the fuel and oxidant channels are parallel, which accommodates co- and counter-flow configura- tions. Cross-flow configuration is another option for flow arrangement in state-of- the-art SOFC manifolding. 13 Oxygen is oxidized in the cathode by electrons coming from the anode via the exter- nal circuit. Oxygen ions transport through the electrolyte to the anode where they combine with hydrogen and/or carbon monoxide to produce water and/or carbon dioxide and release electrons. The interconnects carry electrons from electrochemi- cal reaction sites to the external circuit on the anode side and do the reverse on the cathode side. When stacking cells with planar geometry in series or parallel, they function as electrical connections between neighboring cells and as gas separators. SOFCs are facing challenges which need to be solved due to the high operating temperature. High thermal stress in the fuel cells or fuel cell systems, for instance, accelerates the material degradation processes and has been shown to be the main cause of cell component breakages. Therefore, further research aiming at under- standing the detailed processes or phenomena happening in the cells is needed. Amongst those, an accurate numerical approach, which enables modeling detailed physical and chemical processes and hence works as a numerical tool for optimizing cell performance, is the aim of this thesis. 2.1.1 Chemical and electrochemical reactions Chemical Reactions The high operating temperature (500–1000◦ C) of solid oxide fuel cells makes it pos- sible for the cells to work directly with hydrocarbon fuels, reducing the need for a complex and expensive external fuel reforming. This is impossible for the low- and medium-temperature fuel cells. It is common to use natural gas as fuel. With presence of nickel metal (Ni) in the anode, methane is strongly reformed producing CO and H2 . CO is then further transformed into CO2 via a shift reaction. The reforming and shift reactions are described as CH4 + H2 O ­ CO + 3H2 , (2.1) CO + H2 O ­ CO2 + H2 . (2.2) The reforming reaction is endothermic, therefore consuming heat generated by the exothermic electrochemical processes. The Boudouard reaction: 2CO ­ CO2 + C and the cracking reaction: CH4 ­ C + 2H2 are the main pathways for carbon formation in the anode at high tem- perature. This is a serious issue as carbon tars block active sites for chemical and 14 electrochemical processes and impede transport of the gas phase, therefore reducing cell performance. However, addition of excess steam to the fuel stream shifts the reactions away from carbon formation [7–9]. For SOFCs working with natural gas, steam-to-carbon ratios of 2.5–3 are common. Electrochemical Reactions In a typical SOFC, electrochemical reactions take place at three-phase boundaries (TPBs) formed by the presence of all three of the ionic phase, the electronic phase and the gas phase. These electrochemically active sites are mostly located at electrode–electrolyte interfaces. However, in case composite electrodes are used, the active sites can extend further into the electrodes up to a dept of 50µm [10–13]. Reduction of oxygen on the cathode side, and oxidation of hydrogen and carbon monoxide on the anode side are described, respectively, as 1 O2 + 2e− ­ O2− , (2.3) 2 H2 + O2− ­ H2 O + 2e− , (2.4) CO + O2− ­ CO2 + 2e− . (2.5) Overall cell reactions are therefore 1 H2 + O2 = H2 O, (2.6) 2 1 CO + O2 = CO2 . (2.7) 2 The overall cell reactions are exothermic, maintain the cell at the high temperatures required for reasonably high ionic conductivity of the electrolyte and reaction rates at the TPBs. Actually, CO mostly participates in the water-gas shift reaction of Eq. (2.2) rather than in the electrochemical processes [14]. In a system where H2 and CO coexist, the rate of CO oxidation is around 2–3 times less than that of H2 oxidation depending on the operating temperature [15]. 2.1.2 The electrolyte The SOFC electrolyte is a ceramic material sandwiched by the anode and cathode. The electrolyte functions as an ionic conductor enabling oxygen ions to flow from the three-phase boundaries on the cathode side to those on the anode side through its 15 crystal lattice. Moreover, with its dense solid nature, it also works as a gas separator, preventing gas species from penetrating into it. Additionally, the electrolyte can function as a mechanical supporting structure with thickness 100–200µm, i.e. in electrolyte-supported SOFCs [3, 16, 17]. However, more and more attention is given to electrode-supported designs with a very thin electrolyte of 5–20µm [16–18]. A thin electrolyte reduces ohmic resistance to ion transport in the electrolyte. An SOFC electrolyte material must meet various requirements in order for the fuel cell to have a good performance and be stable over long time of operation. These include [1, 2, 9, 19, 20]: – high ionic conductivity, – negligible electronic conductivity, – chemical stability in both reducing and oxidizing environments, – thermodynamic stability over a wide range of temperature and oxygen partial pressure, – thermal expansion compatibility with materials of electrodes and of other com- ponents, e.g. interconnects, sealants. Common materials of SOFC electrolyte are zirconia, ceria fluorites and LaGaO3 - based perovskites [19]. Yttria-stabilized zirconia (YSZ) is currently the most commonly used material for SOFC electrolytes working at temperatures higher than 700◦ C since it fulfills the necessary requirements. Scandia-stabilized zirconia (ScSZ) has higher ionic conductivity than the conven- tional YSZ material [21, 22]. However, a drawback of ScSZ is performance degrada- tion over long-term exposure to high temperatures. Therefore, this type of material is mainly attractive for intermediate-temperature (600–800◦ C) SOFCs [19, 23, 24]. Doped-CeO2 electrolytes, e.g. gadolinium doped ceria (GDC) are only attractive for low-temperature (< 600◦ C) SOFCs since they are partially reduced in hydrogen at temperatures above 600◦ C [3]. LaGaO3 -based electrolytes, typical lanthanum strontium gallate magnesite (LSGM), show high ionic conductivity and can be used for intermediate-temperature SOFCs. However, challenges remain in matching the thermal expansion coefficients, mechan- ical strength and chemical compatibilities [3]. 16 2.1.3 The electrodes The electrodes are in principle electronic conductors, forming together with the elec- trolyte and the gas phase the three-phase boundaries (TPBs) where electrochemical reactions take place. Therefore, they must be porous to allow gas species to transport in and out the TPBs. The electrodes also catalyze the electrochemical reactions. However, electrode materials have to fulfill a number of conditions because of the high operating temperature. The anode material has to be chemically, morphologi- cally, and dimensionally stable in the fuel gas environment, and likewise the cathode material in the air environment during cell operation. Moreover, the anode material has to be tolerant toward contaminants possibly available in fuel stream. Other conditions of the electrode materials include [3, 25–29]: – high electronic conductivity, – sufficient porosity to facilitate transport of reactants and/or products to and/or from the TPBs, – chemically, thermally and mechanically compatibility with other cell compo- nent materials during fabrication as well as under operation. Nickel can be used as anode material since nickel metal plays the dual role of hydro- gen oxidation catalyst and electric current conductor. Additionally, nickel is also an excellent catalyst for cracking of hydrocarbons, e.g. in situ reforming of methane. However, the thermal expansion of nickel is considerably higher than that of the yttria-stabilized zirconia (YSZ) conventionally used for the electrolyte. Another problem with nickel is that it can sinter at the cell operating temperature, causing decreasing porosity and reduction of the TPB [26]. Strontium-doped lanthanum manganite (LSM) is the most widely used material for the cathode. Composite electrodes made of a binary mixture of electronically and ionically con- ducting particles are more and more widely used in state-of-the-art SOFCs since they are superior to electrically conducting electrodes. An advantage of the composite electrodes is that the TPBs can extend into the electrodes, resulting in reduction of activation losses associated with the electrochemical processes. Figure 2.3 repre- sents the TPBs in electrodes, which are made of electronically conducting, a binary 17 mixture of electronically and ionically conducting, and mixed-conducting particles, respectively [30]. As will become evident, cases b) and c) cannot be distinguished in the model developed in this thesis and they will both be referred to as ”mixed- conducting electrodes”. With mixed-conducting electrodes, the TPBs do, as the figure shows, extend into the electrodes from the electrode–electrolyte interfaces. Electronically / Ionically / Mixed/ conducting / conducting / conducting / particles particles particles Electrolyte Electrolyte Electrolyte a) b) c) Figure 2.3: TPBs (arrowed) in a) electronically conducting, b) composite and c) mixed-conducting electrodes. Common composite electrodes are Ni–YSZ and LSM–YSZ for the anode and cath- ode, respectively. Other advantages of the composite electrodes include: – reduction of mis-matching of the thermal expansion: the thermal expansion coefficient of YSZ is closer to that of Ni–YSZ mixture than to that of pure Ni [5]. This also allows better anode–electrolyte adhesion; – prevention of nickel sintering: the presence of YSZ particles between Ni parti- cles in the Ni–YSZ mixture prevents agglomeration of the metal particles. CeO2 based materials, e.g. doped with Gd, Sm and Y, are typical mixed-conducting anodes [28, 31, 32]. Perovskite materials such as Sr-doped LaCoO3 (LSC), LaCoO3 co-doped with Sr and Fe (LSCF) are examples of mixed-conducting cathodes. These cathode materials are suitable for SOFCs operating at intermediate and low tem- peratures. An electrode can be a mechanical supporting structure in a fuel cell, in which case it is the thickest component compared to that of the electrolyte and the other electrode; this is an electrode-supported cell. In anode-supported cells, the anode thickness is 0.5–1.5mm, while in cathode-supported cells, the cathode thickness is 0.3–1mm. The other electrode thickness is ∼50µm [3, 17]. 18 2.1.4 The interconnect The interconnect transports electrons between the electrochemically active sites (TPBs) and the external circuit. In a typical SOFC, the interconnect is in direct con- tact with both the anode and cathode and both the fuel and air. Therefore, require- ments for interconnects are most severe of all cell components, namely [3, 33–36]: – good electrical conductivity, – chemical stability in both oxidizing and reducing environments at the cathode and anode, respectively, at high operating temperatures, – chemical stability with other cell components during cell operation and fabri- cation, – dimensional stability with changes in temperature and/or oxygen partial pres- sure, – thermal expansion matching that of the other cell components, – low permeability for oxygen and hydrogen (or fuel) minimizing their direct combination during cell operation, e.g. in planar geometrical designs, – adequate mechanical strength. There are two types of materials for state-of-the-art SOFC interconnects, namely ceramic and metallic, with different features. The ceramic lanthanum chromite is the most common material for SOFC inter- connects working at high temperatures (900–1000◦ C) since it is stable in oxidizing environments at the cathode. Metallic interconnects have a better electrical conductivity compared to ceramic ones, but are not stable in oxidizing conditions. Therefore, they are mainly suitable for lower temperatures [35]. Oxidation resistant alloys based on Cr or Ni are suitable for intermediate-high temperatures (800–900◦ C). For SOFCs working at intermedi- ate temperatures (650–800◦ C) ferritic stainless steel is more favorable [3, 33]. Moreover, metallic interconnects have an interesting feature, which is of mechanical strength. Therefore, they can be used as mechanical support in planar SOFCs, in so-called interconnect-supported cells. This makes it possible to use thin electrolytes (5–15µm) and electrodes (∼50µm), reducing ohmic losses considerably, and hence increasing cell performance. 19 2.2 Aim of the current study The main aim of this work is to develop a numerical model for solid oxide fuel cells (SOFCs). This model can be used to study detailed phenomena taking place in complex geometries consisting of the gas channels, the porous electrodes, the dense electrolyte and the interconnects. To be able to capture all the detailed processes including mass, heat and charge transports and chemical and electrochemical reactions occurring in the cell, the model should be three-dimensional. A single computational domain covering a whole unit cell will be used in order to avoid problems arising due to manually coupling solutions in separate domains, as in quasi-two or three dimensional models in the literature. In this single computational domain, equations describing the detailed processes are therefore resolved simultaneously. Another aim of the thesis is to numerically investigate performance of SOFCs using the developed model. Simulation results can give detailed insights such as distribu- tions of temperature, chemical species, current density and electrical potential in the fuel cells, therefore help optimizing the cell design and performance. Such insights can not be gained by experiments. Different geometries and flow configurations will be investigated. Experimental validation of the model are given where possible in the papers attached to this thesis, though this task is rather difficult because of lack of standardization - insufficient details and/or different data are used in different works in the literature. 20